§1.44 Ὅει δʼ οὐκ ἔστι τῇ κύκλῳ φορᾷ ἐναντία ἄλλη φορά, πλεοναχόθεν ἄν τις λάβοι τὴν πίστιν, πρῶτον μὲν ὅτι τῇ περιφερεῖ τὴν εὐθεῖαν ἀντικεῖσθαι μάλιστα τίθεμεν.
4 That there is no other movement contrary to circular movement may be believed on several grounds. First, because we postulate that the straight line is most opposed to the circular.
τὸ γὰρ κοῖλον καὶ τὸ κυρτὸν οὐ μόνον ἀλλήλοις ἀντικεῖσθαι δοκεῖ ἀλλὰ καὶ τῷ εὐθεῖ, συνδυαζόμενα καὶ λαβόντα σύνθεσινʼ ὥστʼ εἴπερ ἐναντία τίς ἐστι, τὴν ἐπὶ τῆς εὐθείας μάλιστα ἀναγκαῖον ἐναντίαν εἶναι πρὸς τὴν κύκλῳ κίνησιν.
For the concave and the convex seem to be opposed not only to each other but also to the straight, when combined and taking a composite form; so that if there is any contrary movement, the movement along the straight line must of necessity be most contrary to the circular movement.
αἱ δʼ ἐπὶ τῆς εὐθείας ἀλλήλαις ἀντίκεινται διὰ τοὺς τόπους· τὸ γὰρ ἄνω καὶ κάτω τόπου τέ ἐστι διαφορὰ καὶ ἐναντίωσις.
But movements along the straight line are opposed to each other because of their places; for upward and downward are a difference and contrariety of place.
ἐπείτ’ εἴ τις ὑπολαμβάνει τὸν αὐτὸν εἶναι λόγον ὅνπερ ὑτὶ τῆς εὐθείας, καὶ ἐπὶ τῆς περιφεροῦς (τὴν γὰρ ἀπὸ τοῦ Α πρὸς τὸ Β φορὰν ἐναντίαν εἶναι τῇ ἀπὸ τοῦ Β πρὸς τό Α), τὴν ἐπὶ τῆς εὐθείας λέγει· αὕτη γὰρ πεπέρανται, περιφερεῖς δʼ ἄπειροι ἂν εἶεν περὶ τὰ αὐτὰ σημεῖα.
Next, if anyone assumes that the same reasoning holds for the circular as for the straight line (namely, that the movement from A to B is contrary to that from B to A), he is speaking of the movement along the straight line; for this latter is finite, while there could be infinitely many circular lines about the same points.
ὅμοίως δὲ καὶ ἐπὶ τοῦ ἡμικυκλίου τοῦ ἑνός, οἷον ἀπὸ τοῦ Γ ἐπὶ τὸ Δ καὶ ἀπὸ τοῦ Δ ἐπὶ τὸ Γ· ἡ γὰρ αὐτὴ τῇ ἐπὶ τῆς διαμέτρον ἐστίν· ἀεὶ γὰρ ἕκαστον ἀπέχειν τὴν εὐθεῖαν τίθεμεν.
Similarly also on a single semicircle, for example from C to D and from D to C; for this is the same as the movement along the diameter, since we always assume that each point is at a straight line distance.
ὁμοίως δὲ κἂν εἴ τις κύκλον ποιήσας τὴν ἐπὶ θατέρου μικυκλίου φορὰν ἐναντίαν θείη τῇ ἐπὶ θατέρου, οἷον ἐν τῷ ὅλῳ κύκλῳ τὴν ἀπὸ τοῦ Σ πρὸς τὸ E τοῦ H ἡμικυκλίου τῇ ἀπὸ τοῦ Ζ πρὸς τὸ E ἐν τῷ θ ἡμικυκλίῳ.
Similarly, even if someone made a circle and assumed that the movement on one semicircle was contrary to that on the other, for example, in the whole circle, the movement from Σ to E on the H semicircle was contrary to that from Z to E on the θ semicircle.
εἰ δὲ καὶ αὗται ἐναντίαι, ἀλλʼ οὔτι γε αἱ ἐπὶ τοῦ ὅλου κύκλου φοραὶ ἀλλήλαις διὰ τοῦτο ἐναντίαι.
Even if these are contrary, yet the movements on the whole circle are by no means contrary to each other because of this.
ἀλλὰ μὴν οὐδʼ ἡ ἁπὸ τοῦ Α ἐπὶ τὸ Β κύκλῳ φορὰ ἐναντία τῇ ἀπὸ τοῦ Α ἐπὶ τὸ Γ· ἐκ ταὐτοῦ γὰρ εἰς ταὐτὸ ἡ κίνησις, ἡ δʼ ἐναντία διωρίσθη φορὰ ἐκ τοῦ ἐναντίου εἰς τὸ ἐναντίον.
Furthermore, the circular movement from A to B is not contrary to that from A to C; for the movement is from the same to the same, whereas contrary movement is defined as being from the contrary to the contrary.
εἰ δὲ καὶ ἦν ἡ κύκλῳ τῇ κύκλῳ ἐναντία, μάτην ἄν ἦν ἢ ἑτέρα· ἐπὶ τὸ αὐτὸ γὰρ, ὅτι ἀνάγκη τὸ κύκλῳ φερόμενον ὑποθενοῦν ἀρξάμενον εἰς πάντας ὁμοίως ἀφικνεῖσθαι τούς ἐναντίους τόπους·
And even if there were a circular movement contrary to the circular, either one would be in vain or the other; for they are toward the same, because it is necessary that what is carried in a circle, starting from anywhere, reaches all the contrary places alike.
εἰσὶ δὲ τόπου ἐναντιότητες τὸ ἄνω καὶ κάτω καὶ τὸ πρόσθιον καὶ ὀπίσθιον καὶ τὸ δεξιὸν καὶ. ἀριστερόν.
And the contrarieties of place are upward and downward, front and back, right and left.
αἱ δὲ τῆς φορὰς ἐναντιώσεις κατὰ τὰς τῶν τόπων εἰσὶν ἐναντιώσεις·
And the contrarieties of movement are according to the contrarieties of places.
εἰ μὲν ἄρʼ ἴσαι ἠσαν, οὐκ ἂν ἦν κίνησις αὐτῶν, εἰ δʼ ὁ ἑτέρα κίνησις ἐκράτει, ἑτέρα οὐκ ἂν ἦν· ὥστʼ εἰ ἀμφότερα ἦν, μάτην ἄν θάτερον ἦν σῶμα μὴ κινούμενον τὴν αὑτοῦ κίνησιν· μάτην γὰρ ὑπόδημα τοῦτο λέγομεν, οὖ μή ἐστιν ὑπόδεσις.
If, therefore, they were equal, there would be no movement of them, and if the one movement prevailed, the other would not be; so that if both were, one of them would be in vain, a body not moving with its own movement; for we call that shoe in vain of which there is no wearing.
ὁ δὲ θεὸς καὶ ἡ φύσις οὐδὲν μάτην ποιοῦσιν.
But God and nature do nothing in vain.